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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/| i p c / 36 2 ■) . ELEMENTS OP . Infinitesimal Calculus. BT JOSEPH BAYMA, S. J., Prqfemr of Ma£h£iMU^ in Santa CUxra OoUege, 8,J,, Santa Clara, Califomia. San Fkancisoo : A. WALDTEUFEL, 787 MARKET STREET. \ COPTRIGHT, 1889, By a. WALDTEUFBL. w "^ ^ > > PREFACE The present treatise, as its title points out, has been based on the principles of the infinitesimal -^ method. I think that this metliod, besides being ^ the simplest of all for teaching Calculus, as all nf admit, is also the most consistent and philoso- \ phical. Some modern writers, indeed, have tried r< to discredit it ; but, when we examine their reason- ^ ings, we soon discover that they have done so be- cause they have failed to grasp the true nature of infinitesimal quantities. This I have endeavored to show in mv introduction to this work, where the reader will find not only what I believe to be the exact notion of the infinitesimal, but also a hint at the grounds (elsewhere developed) upon wliich the infinitesimal method rests its claim to be pre- ferred to its fashionable rival, the method of limits. This work being intended for young men who are supposed to devote a considerable part of their time to the study of mental and of natural philoso- phy, it has been necessary to limit its developments by giving less prominence to the analytical than to the practical portion of it. But, while anxious not PREFACE. to overtax our young people, I have nevertheless collected and condensed all that seemed to be of practical use in this branch of study; and, though I have made it a point to be concise, I have constantly endeavored to make all things as plain as the subject matter permitted. I hope that the average student will need only a mode- rate effort to understand the object and the pro- cesses of differentiation and of integration as laid down in this treatise ; though, as to integration, his success will often depend also on his endurance of analytical work. To assist him in the performance of this task, I have developed a sufficient number of geometrical and of mechanical problems, to- gether with some fundamental notions of rational mechanics, which were indispensable, and which will serve as an introduction to the regular study of this latter science. J. B. ^ "^■^ I CONTENTS. PAes Inteoduction 9 PART I.— DIFFERENTIAL CALCULUS. Section I. Mules of Differentiation, Algebraic functions of one yai-iable, 21 Transcendental functions of one variablei .... 27 Functions of two or more variables, 36 Implicit functions, 40 Section II. Successive Differeyitials* Maclaurin*s formula, 46 Taylor's formula, 49 De Moivre's formulas, 55 Maxima and minima, 56 Exercises on maxima and minima, 63 Values of functions which assume an indeterminate form, . 79 Section III. Investigatiom about Plane Curves, Tangents, normals, etc., . 85 Direction of curvature .90 Singular points, 93 Order of contact, osculation, 101 Measure of curvature, 104 Evolutes, .108 Envelopes, . . , . . ... . . .113 Elements of arcs, surfaces, and volumes, . . • .117 5 CONTENTS, Differentials with polar co-ordintftes, Spirals, PAOB 119 PART U.— INTEGRAL CALCULUS. Section I. Various Methods for Finding Iniegrcds Integration of elementary forms, , Reduction of differentials to an elementary form Integration by parts, Integration of rational fractions, Integration of binomial differentials. Integration by successive reduction, Integration of some trinomial differentials, Integration by series, Integration of trigonometric expressions, . Integration of logarithmic differentials. Integration of exponential differentials, Integration of total differentials of the first order. Integration of tlie equation Mdx -h Ndy = 0, Integration of other differential equations, Integration by elimination of differentials. Double integrals, Section II. Application of Integral Calculus to Geometry Rectification of curves, Quadrature of curves, . Surfaces of revolution, Solids of revolution, Other geometrical problems, Problems solved by double or triple integrals. Section III. Application of Integral Calculus to Mechanics Work, MoTement uniformly varied, Movement not uniformly varied, 132 138 142 144 151 158 161 168 165 169 170 172 175 182 192 197 199 204 208 210 213 223 . 231 . 287 . 238 CONTENTS. 7 PAGE Composition and decomposition of forces 241 Moments, 245 Virtual moments, 249 Attraction of a sphere on a material point, . . . 251 Centre of gravity, 256 Moment of inertia, 268 Curvilinear movement, 272 INFINITESIMAL CALCULUS. INTRODUCTION. 1. The Infinitesimal Calculus is exclusively con- cerned with continuous quantities ; for these alone admit of infinitesimal variations. A variable quantity is said to be continuous^ when it is of such a nature that it cannot pass from one value to another without passing through all the intermedi- ate values. All the parts of a continuous quantity are continuous : and, as all continuum is divisible, every part of a (Continuous quantity, how small so- ever it be, is still further divisible. In other terms, the division of continuum can have no end. 2. Infinitesimal quantities are sometimes con- ceived as resulting from an endless division of the finite. But this is not the real genesis of infini- tesimals ; for, in the order of nature, it is the infini- tesimal itself that gives origin to the finite. Thus, an infinitesimal instant of duration does not arise from any division of time; for it is the instant itself that by its flowing generates time. In like manner, the infinitesimaj length described by a moving point in one instant of time does not origin- ate in any division of length ; for it is the actual infinitesimal motion of the point itself that by its continuation generates a finite length in space. 9 10 INFINITESIMAL CALCULUS, Hence infinitesimals of time and of length are not mathematical fictions. They are true objective realities. Had they not a real existence in nature, neither the origin nor the variations of continuous movement would be conceivable. For the same reason we must admit that continu- ous action cannot produce acceleration except by communicating at every instant of time an infini- tesimal degree of velocity : and speaking general- ly, all continuous quantities develop hy inflnitesi- mat Tnoments. Hence the branch of Mathematics which investigates the relations between the con- tinuous developments of variable quantities, has re- ceived the name of Infinitesimal Calculus^ and its method of investigation has been called the infini- teHmal method. This method has been used by the best mathe- maticians up to recent times. Poisson, in the intro- duction to his classical Traite de Mecanique (n. 12), says: ^'In this work I shall exclusively use the in- finitesimal metliod. . . . We are necessarily led to the conception of infinitesimals when we consider the successive variations of a magnitude subject to the law of continuity. Thus time increases by de- grees less than any interval that can be assigned, however small it may be. The spaces measured by the various points of a moving body increase also by infinitesimals ; for no point can pass from one position to another without traversing all the iif^ termediate positions, and no distance, how small soever, can be assigned between two consecutive positions. Infinitesimals have, then, a real exist- ence: they are not a mere conception of Mathe- maticians." INFINITESIMAL CALCULUS, 11 3* Modem authors often define the infinitesimal as the limit of a decreasing quantity. This defini- tion we cannot approve. For the divisibility of continuum has no limit, and therefore cannot lead to a limit. This is so true, that even those authors confess that the limit — the absolute zero — can never be reached. On the other hand, infinitesimals, in the order of nature, do not arise from finite quanti- ties: it is, on the contrary, these quantities that arise from them. The origin of infinitesimals is dynamical ; for they essentially either consist in, or depend on, motion: and as motion has no other being than its actual becoming or developing, so also infinitesimals have but the fleeting existence of the instant in which they become actual. It is for this reason that Sir Isaac Newton conceived them 2iS fluxions and nascent quantities ; that is, quanti- ties not yet developed, but in the very act of de- veloping. This is, we believe, the true notion of the infinitesimal, the only one calculated to satisfy a philosophical mind.* So long as it remains true that a line cannot be drawn except by the motion of a point, so long will it remain true that an infini- tesimal line is the jfliuvion ot a point through two consecutive positions. An infinitesimal change may be defined, a change which is brought about in an instant of time. Now, the true instant is the link of two consecutive terms of duration : and it is obvious that between two consecutive terms of duration there is no room for any assignable length. Hence the fleeting in- stant has a duration less than any assignable — ' » ■ ' ' " ' ' * On the modern doctrine and method of limits see the note appended to No. 23. 1 2 INFINITESIMAL CALCUL US, length of duration, that is, it lias a dumtion strictly infinitesimal. And in the same manner, every other infinitesimal change is a link of two consecutive terms ^ or of two consecutive states ; for it takes place in an infinitesimal'instant. 4. But here the question arises : How can an in- finitesimal quantity be intercepted between two consecuiive points? -Consecutive points touch one another and leave no room between them. It would seem, then, that what we call " an infinitesi- mal" is not a quantity, but a mere nothing. We answer that a point in motion has always two con- secutive modes of being in space ; for it is always leaving its last position, and always reaching a fol- lowing position which cannot but be consecutive to the last abandoned.. Now, it is plain, that if the actual passage from the one to the other were not a real change, the whole movement would be without change ; for the whole movement is but a continu- ous passage through consecutive points. But movement without change is a contradiction. It is therefore necessary to concede that between two consecutive points there is room enough for an in- finitesimal change, and accordingly for an infinitesi- mal quantity. As a further explanation of this truth, let us con- ceive two material points moving uniformly, the one with a velocity 1, the other with a velocity 2. Their movement being essentially continuous, there is no single instant in the whole of its duration, in which they do not pass from one point to a con- secutive point, the one with its velocity 1, the other with its velocity 2. But the velocity 2 causes a change twice as great as that due to the velocity 1. INFINITESIMAL CALCULUS. 13 Therefore the ratio of the two movements is, at every instant^ as 2:1. But two absolute nothings cannot be in the ratio 2:1. Therefore the move- ments comprised between two consecutive points are not mere nothings, but are real quantities, though infinitely small. They are, in fact, fluxions^ or nascent quantities^ or, as the Schoolmen would say, quantities in fieri. Nor does it matter that these infinitesimals are sometimes represented by the symbol 0. For this symbol has two meanings in mathematics. When it expresses the result of subtraction, as when we have a--a = 0, it certainly means an absolute n/>thing: but when it expresses the result of di- vision, it is a real quotient^ and it always means a quantity less than any assignable quantity: but because it has no value in comparison with finite quantities, it is treated as a relative nothing y and is represented by 0. Thus, in the equation 00 the zero represents an infinitesimal quantity. This can be easily proved. For it is only continuous quantities that admit of being divided in infinitum : and, when so divided, they give rise to none but continuous quotients, because every part of con- tinuum is necessarily continuous. Now, the ab- solute zero cannot be considered continuous. Therefore the absolute zero c^p never be the quo- tient of an endless division. And in this sense, it is true, as the theory of limits affirms, that a de- creasing quantity may indefinitely tend to the limit zero, but can never reach it. On the other 14 INFINITESIMAL CALCULUS. ' - - .. ^ handy the above equation gives a = Xoo; and this does certainly not mean that the finite quantity a is equal to an infinity of absolute noth- ings. 6. We have said that infinitesimals have no value as compared with finite quantities. A few years agOj an American writer* was bold enough to maintain that this fundamental principle of infini- tesimal analysis is not correct. The principle, •however, has been admitted by the greatest mathe- maticians, and its correctness will not be doubted by any one who understands the real nature of in- finitesimals. The principle, says Poisson (loc. cit.), ''consists in this, that two finite quantities which do not differ from each other except by an infini- tesimal quantity, must be considered as equal ; for between them no inequality ^ how small soever, can be assigned^ ^ ; because the infinitesimal is less than any assignable quantity. Again, it is plain that the infinitesimal is to the finite as the finite is to the infinite. Now, the infi- nite is not modified, as to its value, by the addition of a finite quantity. Therefore the finite is not modi- fied by the addition of an infinitesimal. That the infinite is not modified by the addition of a finite quantity, can be assumed as an evident truth : but it can also be demonstrated. Thus, it is shown in Trigonometry that between the angles J., B, (7 of a * Mr. Albert Taylor Bledeoe in his Philosophy qf Mathematics^ where he (strives to prore that the infinitesimal method should be abandoned. Wo are afraid that philosophical readers will not consider his effort a saccess. INFINITESIMAL CALCULUS, 15 plane triangle there is the relation tan A + tan-B + tan(7= tan A tan jB tan (7/ and this equation, taking J. = 90^ jB=46^ (7 = 45°, gives which shows that the addition of a finite quantity does not modify the value of the infinite. We may draw from Arithmetic a still plainer proof of our principle. Dividing 1 by 3 we obtain i = 0.333333 . . . and multiplying this by 3, we obtain 1 = 0.999999 . . . In this last equation, if the second member be understood to continue without end, the difference between the two members will be an infinitesimal fraction — viz., unity divided by a divisor infinitely great. Now, we can prove, that, notwithstanding this inflnitesirrial difference^ the equation is rigor- ously true. For, let the second member of the equation be represented by x; then 0^ = 0.999999 . . . Multiply both members of this by 10 ; then lOa; = 9.999999 . . . = 9 + a;/ and from this, by reduction, we have This clearly shows that the equation 1=0.999999 . . . is rigorously true. It is plain, therefore, that an infinitesimal difference has no hearing on the value of a finite quantity^ and that no error is com- 16 INFINITESIMAL CALCULUS. mitted by suppressing an infinitesimal by the side of a finite quantity. 6. The notions above developed may suffice as a first introduction to the infinitesimal calculus. We have shown — 1st. That infinitesimals are not nothings, but ob- jective realities : 2d. That infinitesimals are not limits of decreas- ing quantities, but fluxions — that is, quantities in the act of developing, or more briefly, nascent quantities, whose value is less than any assignable value of the same nature : 3d. That infinitesimals may have different rela- tive values, and form different ratios : 4th. That an infinitesimal, whether added to, or taken from, a finite quantity, cannot modify its value. As to the different orders of infinitesimals, of which we slmll have to speak throughout our treat- ise, we have here simply to state the fact, that in- finitely great, and infinitely small quantities are capable of degrees, so that there may be infinites and infinitesimals of different orders, each infinite of a higher order being infinitely greater than the infinite of a lower order, and each infinitesimal of a higher order being infinitely less than the infini- tesimal of a lower order. How this can be, one may not find easy to explain, because both the infinite and the infinitesimal lie beyond the reach of human comprehension : nevertheless we know, not only from Algebra and Geometry, but also from rational philosophy, that such orders of infinites and of infinitesimals cannot be denied. We know that the species ranges infinitely above the indi- INFINITESIMAL CALCULUS. 17 vidaal, and the genns infinitely above the species. Substance extends infinitely less than Being, animal infinitely less than substance, man infinitely less than animal. From this it will be seen that the notion of an infinite infinitely greater than another infinite, is not a dream of our imagination, but a well-founded philosophical conception, familiar to every student of Logic, and admitted, implicitly at least, by every rational being. Let us, then, write the following series : X a? of If we assume 0^=00, it is plain that the first term will be infinitely greater than the second, the second infinitely greater than the third, and so on. The middle term 1 being finite, all the following terms are infinitesimal, and each is infinitely less than the one that precedes it. Hence infinites and infinitesimals are distributed into orders. Thus, if X be an infinite of the first order, a? will be of the second order, a? of the third, etc. ; and in like manner 1 will be an infinitesimal of the first order, X 1 1 -3 of the second order, -^ of the third, etc. 7. The problems whose solution depends on the infinitesimal calculus, are generally such that their conditions cannot be fully expressed in terms of finite quantities. Hence a method had to be found by which to* express such conditions in infinitesimal terms. The part of the Calculus which gives rules for properly determining such infinitesimals and their relations, is called the Differential Calculus. As, however, none of sucli 18 INFINITESIMAL CALCULUS, infinitesimals must remain in the final solutions, rules were also to be found for passing from the infinitesimal terms to the finite quantities, of which they are the elements ; and to effect this, a second part of Calculus was invented under the name of Integral Calculus. Of these two parts of the infinitesimal calculus we propose to give a substantial outline in the pres- ent treatise : and we shall add a sufficient number of exercises concerning the application of the Cal- culus to the solution of geometric and mechanical questions ; for it is by working on particular ex- amples that the student will be enabled to appreci- ate and utilize the manifold resources of this branch of Mathematics. PART I. DIFFERENTIAL CALCULUS. 8. Our object in this part of our treatise is to find, and to interpret, the relations which may exist between the infinitesimal changes of correlated quantities varying according to any given law of continuous development. Such a law is mathe- matically expressed by an equation between the variable quantities ; and it is, therefore, from some such equation that the relative values of the infini- tesimal changes must be derived. An infinitesimal change is usually called a differ- ential^ because it is the difference between two con- secutive values, or states, of a variable quantity. The process by which differentials are derived from given equations is called differentiationy and the equations themselves, by the same process, are said to be differentiated. Hence this part of infinitesi- mal analysis received the name of Differeniial Calculus. Differentials are expressed by prefixing the letter d before the quantities to be differentiated. Thus, dx = differential of a?, d{aaf) = differential of oaf. 9. When an equation contains only two vari- ables, arbitrary values can be assigned to one of them, and the equation will give the corresponding 19 20 INFINITESIMAL CALCULUS. values of the other. The one to which arbitrary values are assigned is called the independent vari- able, and the other, whose value depends on the yalue assigned to the first, is said to be a fwfiction of the same. Thus in the equation of the parabola, 2/* = 2pa?, if we take x as independent, y will be a function of x. When an equation contains more than two vari- ables, then all the variables but one can receive arbitrary values, and are, therefore, independent, whilst the remaining one will be a function of all the others. Functions are often designated as follows : y=f{^\ z = ^(^, y\
and
_ 1
y—\/s=^s^,
the diflEerential will be
1 1 I ^ 1 ^-=^ ^ ^ ds ds
If 71=2, then
INFINITESIMAL CALCULUS, 25
that is, the diffei^ential of the square root of a
quantity is equal to the differential of the quantity
divided by twice the radical.
The preceding rules are sufficient for the diflPeren-
tiation of any algebraic function of one variable.
Examples. It is of the utmost importance that
the student should at once familiarize himself with
the above rules of differentiation, and test, by ex-
amples, his practical knowledge of them. Let him
work out the following :
1. y = ax^ -bx-}- ac, dy = (3a^— b) dx,
2. y = {a'+xy-b, dy = \:{a^-\'7?)xdx,
3. 2/ = «^ +- J dy= (^ccx -^)dXy
a + x , 2adx
6. 2/ = 2 ' (!)• (g)' • ■ •
be what y, ^ , -X 5 . • . become in this hypothe-
dx dof
sis. Then
<^' = ^' it) = ^' (S) = ^'^' (g) = ^-^^^ • ■ •
whence
and, substituting these values in (1),
^=<^)+(i)^+(S)f+(S).4
+(S)m+--- (»)
This is Maclaurin's formula. In using it, it is
necessary, of course, that the values attributed to x
be such as will make the series convergent
If a function is not susceptible of development by
this formula, the formula itself will give notice of
the fact ; for, in such a case, some of its constant
factors will become infinite.
EXAMPLES.
1. To develop y = sin or. We have
dy d^y . .
2/ = sin rr, ^ = cosrr, ^ = - sm^r,
d'y_
dx^ '
48 INFINITESIMAL CALCULUS.
1)6I1C6
«=».(i)='.(S)=<>.(S)=-^.--
and substituting in the formula (5),
emx = x- j^ + J 2 3.4.5 ~ 1.2.3.4.5.6:7 + * ' '
This series, being differentiated and then divided
by dx, gives
cos a; - 1 - ^ + 1.2.3.4 1.2.3.4.6.6 + ' • *
2. To develop y = (1 + ^)*- We have
y = {l + xr,^ = n{l+xf-\
g = 7i(7i-l)(?i-2)(l + a;)"-», . . .
II6IIC6
<^)=Mi)=»-(S)=»<'>-»^
and therefore
(1 + xr =
3. To develop y = log (1 + ^)- We have
2/ = log(l + rr),g=-j^,
d^p 1__ fg'y _ 1.2
INFINITESIMAL CALCULUS. 49
hence
.«=»'(i)='.(g)=-.(S)=^.---
and therefore
log(l+:r) = a;~2- + |--^+ . . .
4. To develop y = a*. We have
2^ = «^ S = ^'^ l<>g ^^ S= «" (log «)N
da? ^ ^ da?
^ = a«aogay,
whence
w=''(i)='°«<^(S)=<'°«'">'
and therefore
a» = l+aoga)a. + ^^'^+^^+ . . .
If we make a = ^, whence log a = log ^ = 1, then
we have
Taylofs Formula.
35. Let u^f{x) and t^' =/(;z;+A). Considering
a? and h as two arbitrary parts of a certain line, it is
obvious that, if the line receives an infinitesimal in-
crement, the result will be the same, whether the
increment be attached to the part x or to the part h.
In other terms, the result will be the same whetlier
the function i^'=/(^ + ^) be differentiated with
50 INFINITESIMAL CALCULUS.
regard to a:, considering h as constant, or with re-
gard to A, considering x as constant. In the first
case, the differential coefficient of the function will,
be -J— : in the second case it will be -^r- ; and there-
dx dh^
fore we shall have
du' _dml_ ,..
d^^dh'''' ^^
This equation will afford us the means of de-
veloping the function u' =f{x-\'h) into a series ar-
ranged according to the ascending powers of h^
with coefficients that are functions of x alone.
Let us assume a development of the form
u'=zPJ[^Qh-{-RJi' + Sh'-\-TJt'+ ... (2)
in which P^ Q, H^ . . . are functions of x alone.
Differentiating (2) with regard to x, and dividing by
dXy we shall find
dx dx ^ dx ^ dx ^ dx ^ dx ' ^ ^
then, differentiating (2) with regard to 7^, and divid-
ing by dhy
-^' = g + 2i2A + 3Sh' +4Th' + . . . (4)
Now, by (1), the first members of (3) and (4) are
equal ; hence their second members are also equal,
and the coefficients of like powers of h in those
second members are equal. Therefore
INFINITESIMAL CALCULUS. 51
But P is the value u of the function when A = ;
and therefore dP = du; hence we shall have
1 eZ^it
These values substituted in the equation (2) give
'^-'^^d^^^d^''/ 2 +^-'2:3
d^u A* ,.
+ ^•2:3:4+ • • • ^^^
This is Taylor's formula. The values attributed to
X and Th must be such as will- render the series crni-
vergent
EXAMPLES.
1. Let u=^af^\ then i^' = (re -|- A)* ; and we have
^ = n (TO - 1) (» - 2) a!»-», . . .
hence
u' = {x + Jif = of" + naf" h -{-''^^^^—^ af"-^ h*
2. Let z^ = log x; then z^' = log (a? + A); and
we have
du 1 d^u 1 ^?^ 2 ^*f^ 2.3
eZa? ic ' rfa?* x"" ' 6?rc" rr' ' dx*^ X'
whence
• •
^
52 INFINITESIMAL CALCULUS.
3. Let u = (f ; then w' = «*** / and we have
^_^ ^-^ ^-^
dx~^' daf~^' da?~^' ' ' '
whence
In this equation make x = and h= 1. Then we
have
_ 111
or
e = 2.718281828469 . . . ,
which is the basis of the hyperbolic logarithms.
4. Let u = BiaXj and u' = sin (a? + y) ; then we
have
du d^u . d*u
-=- = cos Xj '3-5 = — sin X. -5-5 = — cos rr, , . .
dx daf ' dx^ '
whence
y
%
u' = sin (rr + y) = sin rr + y cos rr — ^ sin a?
If we change y into — y, we shall have also
sin (^ — y) = sin re — 2/ cos re — ^ sin x
X y" ' I V • •
• t •
INFINITESIMAL CALCULUS. 53
Scholium. Taylor's formula may be used for the development of a
function u =/(a;, y) of two independent variables.
If we begin by giving to a; an increment h, we shall have
/(«; + *, y) = « + ^A + ^^ + ^^+ . . .
and in this equation, when we give to y an increment k, the first
member will become /(a; -k-h, y -k- k\ and in the second member u
will become
du. dSkk* dht k^
dy ^dy^ %^dy^2.%
j-h will become
dx
=
1
-m'
On the other hand
ds
hence
= f/M+lXf =
whence
a*y = 4m' (a — x).
Differentiating this equation with regard to a, we
find a = — ; and this value substituted in our
y
equation gives
xy = vC ;
and therefore the curve is a hyperbola referred to
its asymptotes.
50. The above theory may also be applied to the
solution of problems in which the pammeter de-
pends on the variation of two quantities. In this
case, the data of the problem must give rise to two
equations. Let us give an example.
The straight line PQ
(Fig. 26) slides between ^
two rectangular axes.
Find the envelope^ or
the locus of its inter sec- q
tions.
For the solution, let
AX, J Fbe the co-ordi-
nate axes. Let AP = a,
AQ = b, PQ = c. Then
the equation of the line a
PQ will be
Fig.te
which, for greater convenience, may be written
thus:
INFINITESIMAL CALCULUS, 117
and in which a and & are subject to the condition
a' + 6* = c\ (2)
Differentiating (1) and (2) with regard to a and 6,
we obtain
^ + 2^ = 0, ada + Ml=0,
whence, by eliminating da and 6^, we shall find
X y X a^y , y Vx
~ = f5, or -=-Tr, andx^ = — r.
Substituting, in succession, these values in (1), we
obtain
y{a' + V) = V and x{a' + V) = a\
or
yc" = &•, and x(f = a* /
whence
a = ojic*, 6 = j/ici
Substituting these values of a and 5 in equation
(1), and reducing, we finally obtain the equation of
the envelope, which is
art -f- yl = c*.
Elements of Arcs. Surfaces y and Volumes.
51. We have seen (No. 33) that if ds is an in-
finitesimal element of the line s referred to rectan-
' gular axes, we have
ds = Vdaf + dy\
118
INFINITESIMAL CALCULUS.
the values of dxvmdi dy being drawn from the equa-
tion of the line. This equation gives the differen-
tial of any arc.
The differential of an area is the infinitesimal area
comprised between two con-
secutive ordinates of the
curve. Thus, if AP and
BQ (Fig. 27) are two con-
secutive ordinates, the area
APQB will be the differen-
tial of the area comprised
between the curve and the ^
axis OX. Now APQB^
ydx-\-\dydx. Hence, denoting APQB by dA^
and neglecting the term i dydx^ we shall have
dA = ydx
as the expression for the differential of a plane area.
If the element PQ of the curve be made to revolve
about tlie axis OX, it will generate an infinitesimal
element of a conical surface. Denote it by dS. Its
expression will be
d8^ 4 12;ry -f 2;: (?/ + dy)\ Vdx' + dy^,
or, reducing.
dS = 27:y Vda^ + dy" ;
and this is the differential of a surface of revolu-
tion.
If the area APQB be made to revolve about the
axis OXy it will generate an infinitesimal element of
a conical volume. Call it d V. Then
dV= * 1 ^y" + ^{y + dyy + 7:y{y + dy) \ dx,
INFINITESIMAL CALCULUS.
119
or, reducing,
and this is the differential of a volume of revohUion.
Bifferenbials with Polar Co-ordinates.
62. Let p=zf{f) be the polar equation of any
curve PL (Fig. 28), the pole being at 0, and OX
being the initial line.
At the point P we have p = OP^ and ^ = POX.
Draw the tangent PB, the normal PC^ and through
the pole draw BO perpendicular to the radius
vector. Then OB will be the subtangent^ and OC
the svbnormal^ while OA perpendicular to the tan-
gent will be the polar distance of the tangent
Let ds = PQ be an infinitesimal element of the
curve. Draw the radius vector OQ^ and, with as
120 INFINITESIMAL CALCULUS.
centre, the arc PR. The infinitesimal triangle PQR
may be considered rectilinear, and right-angled
at R. Hence
P^ = PR + RQ\
But PQ = ds^ PR = pdip^ RQ =dp ; hence, substi-
tuting and extracting the root,
ds = Vdp' + p'df.
Such is the differential of the arc.
Let dA be the differential of the area swept over
by the radius vector. The infinitesimal area POQ
may be considered as the area of a rectilinear tri-
angle having the base OQ and the altitude PR.
Hence
dA^\OQy.PR^\{p-\'dp)pdip,
or, as dp disappears by the side of />,
dA = \ p^dif.
Such is the differential of the area.
Let F denote the angle OPB of the tangent with
the radius vector. This angle and the angle RQP^
may be considered equal, as they differ only infini-
tesimally. Now
therefore
'•■'^«^=i=^'-
ten V=«'
dp
Let OA-p. We have OA-OP sin OP A;
therefore
. ^ ta n Y
j> = /,smT; or ^ = ,^===.
_^-.
INFINITESIMAL CALCULUS,
121
Substituting the value of tan T, and reducing, we
have
p*d(p
Such is the polar distance of the tangent.
Let S, 7^ denote the subtangent. We have
>». r=05= OP. tan y,
or
S.T^p'
dp
Let /S'.iV denote the subnonnal. The proportion
OC: OP:: OP: OB
gives
~" "~ OB Q^dtp^ dip
Let PC^R (Pig.
29) be the radius of
curvature of the line
PL at the point P.
Drawing OM perpen-
dicular to PC pro-
duced, and joining
the points and (7,
we have from the
triangle POC
dp
Fig.f9
OC'^OP'+PC"
--^OP.PCcosOPC. \
But OP cos OPC =
PM=z OA =i?.
Therefore
00'-p' + Jr-2pIl.
122 INFINITESIMAL CALCULUS.
Now, if we pass from P to Q^ the side OC and the
radius M will remain unchanged, p and p alone
being subject to variation. Differentiating, then,
and considering 06^ and R as constants, we lind
= 2pdp — 2Rdp^ whence R = ^-^ •
Spirals.
53. A spiral is a plane curve generated by a
point moving on a straight line while this straight
line is uniformly revolving about a fixed point or
pole. The portion of the curve generated during
one revolution is called a spire. The law according
to which the moving point advances along the re-
volving line determines the nature of the spiral.
Denoting by p the radius vector, and by f the angle
that the radius vector makes with the initial line,
and considering /> as a function of f , the general
equation of a spiral will be
The most remarkable spirals are the spiral of
Archimedes^ the parabolic spiral^ the hyperbolic
spiral^ and the logarithmic spiral.
64. The spiral of Archimedes is generated by a
point moving uniformly along a straight line uni-
formly revolving. If p and p' be two radii vectores,
and tp and ' = 2 t:.
INFINITESIMAL CALCULUS.
123
Tip T
Then p = ^^ or, making for greater simplicity 5-
= a.
27t
p = a(p.
Stt
(1)
Sucli is the equation of the
spiral of Archimedes. Its
form is the same as that of
the equation y = dx of a
straight line passing through
the origin of co-ordinates.
Tills spiral may be con-
structed by dividing the cir-
cumference into a number of
equal parts, and the radius
into the same number of
equal parts, and then taking
as many parts of r for the
radius vector as there are
corresponding parts taken
on the circumference.
Differentiating (1) we have
dp = adif Substituting this
in the general formulas of
No. 52, we shall find
ds = adip Vl + f •,
dA = ^cCif^dif^
tanF= ^ =fl>,
p= — ^
Vl + y'
72 = ^(1+?!)',
2 + 9^"
Fig, so
124 INFINITESIMAL CALCULUS.
It will be remarked that the subtangent CB cor-
responding to f = 2;r is equal to the circumference
of the measuring circle ; for ay" becomes a(2r)"
= jj^ (2;r)* = 2;rr. Hence the area of the triangle
ABC is ir X 2;rr = ;rr* = area of the circle.
55. The parabolic spiral (Pig. 31) is so called be-
cause its equation
/>• = 2af (2)
is of the same form as the equation y* = 2ax of the
parabola. It may be con-
structed by first constructing !
the parabola and then laying
oflf from A to D along the /
circumference of a measnr- /
ing circle any assumed ab- I
scissa Xy and drawing from \
the centre C towards D \
the corresponding ordinate
y as a radius vector. Let r
be the radius of the measuring circle ; then - be-
comes y, and y becomes p. Hence after a revolution
we shall have - = 2;r, and y = CA = r, and the
equation (2) will give r"= 2a X 2n-, or a = r-.
Diflferentiating (2), we have pdp = ad we find w = A- Therefore
tan-' -A = tan-» i + tan-' A-
Assuming ^ = A, m = i, we find n--T^. There-
fore
tan-* A = tan-* i — tan-' ^.
Hence, by successive substitutions, we at last find
tan"^ = - = 4 tan"^ \ — tan"^ ^^.
Thus, if we make x^\\n the formula, we find
the value of tan-^, and in like manner, if we make
0? = ^T we find the value of tan'^ -^^ The calcula-
tion of six terms for tan-*|, and of two terms for
tan"^ irfr? will give the value of ^ np to the tenth
decimal.
91. The differential of an arc, when expressed in
function of polar co-ordinates, is (No. 52)
#
ds ~ Vcip"" + p'dip\
p being the radius vector, and (p the angle which p
makes with the initial line. The integration will
be made as follows.
INFINITESIMAL CALCULUS, 203
Example I. To find an arc of the spiral of
Archimedes. The equation of the curve is /> = ay,
where a — ^r- Hence d(p = — . and
Integrating by i)arts,
and the last term, integrated by formula (33), gives
Hence
^ = h,P ^"^P^ +1 log 0* + V7+T') + C;
and taking Hie integral from p = to p = r, we
have for the first spire
or, since a = ^r- ,
^ = 1 1/1 + 4? + 1^ log (2;r + vT+i?).
To obtain the length of n spires, it suffices to
take the integral from /t> = to /> = nr.
Example II. To find the length of the curve
traced by the end ^ of a tense string AB (Fig. 33),
whose other end A is fixed on a circle around which
the string is being wrapped.
i
1
Ftflf.W
204 INFINITESIMAL CALCULUS,
Let AB = ?, and AC-^r^ and let ^ be a point
on the curve. Draw ED tangent to the circle, and
draw the radius CD.
Make ACD^^, and b
consequently, the arc
AD = r^. Then the in-
finitesimal arc ds = EF
described by ED will be
equal to ED X d&. But,
by the nature of the
case, ED ^AB — AD ^l-^r^. Therefore
Integrating from i> = to i> = i?, we have
As a particular case, assume I = 27tr ; then the
whole curve will be described when * = 2n. Then
*=2;r'r,or^ = 2j..
Quadraiure of Curves.
92. The differential of the area of a curve re-
ferred to rectangular axes is (No. 51)
dA = ydx.
This equation, integrated between proper limits,
will give us the area intercepted between the curve,
the axis of rr, and the limiting ordinates.
Example I. Let the curve be a x>arabola y' = 2px.
Then dx = ^-^ ; whence
P
dA = t^
P
INFINITESIMAL CALCULUS, 205
Integrating from y = Otoy=^yy we have
Example II. Let the curve be a cycloid. Then
dx==— —=====. whence
v2ry — y*
rfl ^^^
V 2ry — y*
Integrating by formula (37), we have
f V'dy ^ _ p V2ry - y' , 3r / " y% .
y \f2ry-y^ 2 "'' 2y V^ry — y'
and again, by tlie same formula,
f—^^- = - 1/27-2/ - «/• + r vers-' ^ •
I
Substituting this value in the last term of the pre-
ceding equation, and taking the integral from y =
to y = 2r, we have
-A = TT- vers"^ — = -^r- •
2 r 2
This is the area of a semi- cycloid. The area of the
whole cycloid is therefore 3;rr', or three times the
area of the generating circle.
Example III. Let the curve be a circle. In this
case, since y = Vr^ — af , we have
dA = dx Vr' — a;'
whence (No. 73)
206 INFINITESIMAL CALCULUS.
Taking this integral from a; = to a; = r, we shall
have
A = \nr\
for a quadrant, and therefore 4-4 = ;rr' for the
whole circle.
Example IV. Let the curve be the logarithmic
y = log X. Then
dA = log xdx.
Integmting by parts, we have
/ log xdx = X log X— / dx = x (log a: — 1) -+- (7.
Taking this integral from ^ = 1 to a: =x , we have
A =x log a? — re + 1^
for the area extending above the axis of re, so long
as we take x>l. But if we take the integral from
a; = to re = 1, then we find
^ = l~[rr(loga:-l)]o = l;
for we know (No. 37) that x log a? - a? = when
x=0. Thus A = lis the area extending to infin-
ity beneath the axis of x from a? = 1 to a? = 0, that
is, from y=0tO2^=— c».
Example V. Let the curve be the equilateral
hyperbola xy = m". Then y = — ; hence
X
a A =zm — ,
X
and integluting from a? = 1 to a? = a?,
A=m*logx,
INFINITESIMAL CALCULUS, 207
When m = 1, then A = log x. Thus the Napierian
logarithms, whose modulus is = 1, exhibit so many
areas taken in the hyperbola xy = l. Hence they
are also called hyperbolic logarithms.
93. When the curve is referred to polar coordi-
nates, then (No 52)
dA =z ^p'dip.
In the spiral of Archimedes^ in which p = af ,
and dp = a^f , we have
dA = \(^ip^dAp^ or dA = ^ p^dp.
Integrating the first expression from ^ = to
^ = 2;r, and the second from* ^ = to /> = r, we
have
^ = I' (2;ry, and ^ = |.^';
r
and, as a = ^ , both expressions reduce to -4 = J;rr'.
In the parabolic spiral^ in which />' = 2af , and
pdp = adipj we have
dA =a(pdf^
and integrating from y = to ^ = 2;r,
A = 2a7r\
r*
But, as a = J- (No. 56), this surface becomes
47r
208 INFINITESIMAL CALCULUS,
In the hyperbolic spiral^ in which pip = a, and
di(>= r » we have
dA = ^^p' -Y = -^^P'
And integrating from p=pto p^a^
a being the radius of the measuring circle.
In the logarithmic spiral^ in which f = a log />,
and d
AP z z
where a = CP. On the other hand in the triangle
PAC we have
AP' = s' = a*+r'- 2ar cos ^CD = a' + r* - 2ax;
therefore
m (a — x) dx
^"^ 2f 4/ (a- + r" -"2a^' '
or
^ m \^ adx xdx )
^~2r I t/(a^-j-r«_"2^"" V (a^+T^-2axf \ *
Now
adx 1
J Via
(a' + r' - 2axf V a" + r' — 2ax
INFINITESIMAL CALCULUS.
25»
and
icdct
V{a''\-r''-2uxf
X
Hence, substituting, and reducing,
-4- ! •
^ a'
9
rri
2r
ax — r
+ ^;
a* V'a' + r' - 'lax
and taking the integral from x^=^ — r to a; = r, and
reducing,
__ m j r {a - r) r {a -\-r) ) __ w .
f
a
a-\-r
that is, the total action is the same as if the whole
mass of the shell were concentrated in C. The same
being true of all the shells into which a solid sphere
may be decomposed, it follows that the action of a
sphere formed of homogene-
ous shells is the same as if its
mass were concentrated in its
centre.
Fig.J^
109. If the point P were
placed anywhere within the
spherical shell (Pig. 45), the
resultant of all the actions of
the shell upon it would con-
stantly be ==5 0, and therefore the point P wonld
remain in equilibrium. For, in this case, the dif-
ferential equation would be
{x — a) dx
cUp =
m
2r
i^{a' + r'-2axy
254 INFINITESIMAL 'CALCULUS.
which differs only by tlie sign from the equation
of the former case. Hence the integral will be
m r* — ax , ^
and this, if taken from a? = — r to a? = r, will give
_ m i r {r — a) r (r -{-a) ) __
^ ~ 2ra* ( r —a r-^a f""'
And this is true of all the shells whose radius is
greater than CP. Accordingly, if the point P were
placed within a solid sphere, it would be attracted
as if the shells beyond CP had no existence ; that
is, it would only feel the attraction of the nucleus
PQH.
Corollary. If an opening were made along one of
the diameters of the earth, and a body allowed to
fall through it, the body (abstraction being made
from the resistance of the air) would be urged to-
wards the centre by an action varying as the simple
distance from the centre.
For, let r be the radius, and p the density (sup-
posed uniform) of the earth. Its mass will then be
47rr*
-^ p\ whilst the mass of the nucleus PQR will
be -^— />, s being its radius. Now, the action of
the earth at its surface is ^ = -^ p —^ , and the
action of the nucleus at a distance s from the
centre is g^ = -^- p -^ ; whence
4
g' :g::s:r, or g' = g -•
INFINITESIMAL CALCULUS. 255
Hence the equation for the movement of a point
approaching the centre, is
d^8 _ gs
Multiplying by 2ds, and integrating, we have
(§)■= - •? + «
Making ^ = when s = Ty we have C =i — ^
Therefore
(§)■= '•=?('■- *•)• (1)
When the body reaches the centre, then 5 = 0^
and v=- Vgi\ which is the maxhnum velocity.
When the body has reached the centre, its velocity
will carry it further on, and, as s changes its sign,
the motion will be retarded instead of accelerated,
until V reduces to zero when ^= — r. Then the
body will fall again towards the centre, and meas-
ure backward the same diameter, and perform a
continuous series of oscillations of the same kind.
Extracting the root of equation (1) and taking
the radical negatively, because ds and dt have op-
posite signs, we find
ds
dt
-^.-
and this integrated from * — r to s=: — r, gives
t = 7T A/-.
y g
This is the time of one entire excursion. This time
256 INFINITESIMAL CALCULUS.
is equal to that in which a body would measure the
semi-circumference rr witli a uniform velocity
= VyT ; for, if t Vgr = ;rr, then ^ = ;r A/ - • -
The time t of the excursion is independent of the
distance from which the body begins to fall. For,
since we have
r:g::s:g\
we can replace the radical A/ - by 4/ -, , without
altering the value of t. Hence all the excursions
will be isochronous, whatever may be their iimpli-
tude. But these results would be greatly modified
by the resistance of the air, which we have
neglected.
Centre of Gravity,
110. The centre of gravity of a body is a point
within the body, through which the resultant of
the actions of gravity on each particle of the body
always passes. All these actions are directed
towards the centre of the earth ; yet they may,
without error, be considered parallel. Hence their
resultant is their sum (No. 105).
It is obvious that the centre of gravity of a
straight line is at its middle point ; also, that the
centre of gravity of a plane figure is in that plane,
and if the figure has a line of symmetry, its centre
of gravity is on that line. In like manner, if a
solid has a plane of symmetry, its centre of gravity
is in that plane.
The centre of gravity of a homogeneous body
INFINITESIMAL CALCULUS.
257
does not depend on the intensity of gravity or ou
the density of the body. Its position depends only
on the form of its volume. We may therefore sub-
stitute volumes for masses and weights, and con-
sider only the relative position of the elements of
which the body is com loosed.
Let M (Fig. 46) be a homogeneous body of any
form. Draw rectangular axes, and let the plane
XY be horizontal. The action of gravity will be
parallel to the axis AZ. Let m be an element of
the body, and let its co-ordinates he x= qn^ y =pny
z = mn. If dv be the
volume of tht^ element ^ ^
m, its moment with re-
spect to the axis AY
will be xdv. Every other
element of the body M
will give a similar mo
ment, the value of x
varying between the
limits of the body. Hence
the sum of the moments
of all the elements with
respect to the axis A Fwill be / xdv.
Ijet now be the centre of gravity of the body,
and Xo = i>(7, y^ = BO^ z^ = 0(7, its co-ordinates.
Since the resultant of the actions of gravity passes
through 0, the moment of the resultant with re-
spect to the axis ^ F will be a?b / dv. Hence, by
the theory of moments (No. 106),
Xq f do = /xdv;
258
INFINITESIMAL CALCULUS.
and therefore
^
fxdo
If the moments were taken with respect to the
axis AX^ we would find in like manner
^0 =
fdr. '
and if the figure were turned about so as to make
the axis of x vertical, we would have also
;^o =
f zdv
7^
Such are the values of the co-ordinates of the
centre of gravity of the body.
111. Centre of gravity of a circular arc. Let
the axis OX (Fig. 4T| bisect the arc
ABC. Then OX will be a line of
symmetry, and the centre of gravity
will lie on OX. Let AC^=c be the
chord, and OA = r the radius of
the circular arc. If be the origin
of co-ordinates, the equation of the o^
circle will be \
and therefore
dv=VM+W=dyA/^=r^ = -^:^.
" " \ 0^ X Vr'—y'
/
INFINITESIMAL CALCULUS. 25^
Substituting in the above expression for a^ we have
frdy
/rdy
and integrating from y = — ic to y=i ^c^
re TO
( . . c .A c\\ arc.A^C*
Making the angle AOX = ??, we have c = 2r sin t?^
whilst the arc ABC^ 2r& ; whence we get also
sirt(?
112. Centre of gravity of a circular segment.
Referring to Fig. 47, where the segment ABC is
bisected by the axis 0X\ we find
dv = 2ydx = ^^ ' - , xdv = 2y^dy:
y r" — 2/'
whence
^^- / - 2y^dy_ '^
J )/r^ ^^
and integrating from 2/ = to y=y^ by formula.
(33),
r^^= ^?^
r'sin-^^ — 2/|/r» — 2/«
2(50 INFINITESIMAL CALCULUS.
and, since y =r sin , substituting and reducing^
we have
_2r sin'
^— 3 • (?-sin«>cos'
11 3, Centre of gramty of a circular sector.
Referring again to Fig. 47, let us conceive that the
sector OABC has been divided into equal infini-
tesimal sectors, every one of which may be looked
U])on as a triangle having the vertex in and an
infinitesimal base on the circumference. The centre
of gravity of every one of these sectors, or triangles,
will be at a distance |r from the centre 0, as can
be shown by a simple geometric construction.
Hence the whole series of these centres of gravity
will determine an arc of circle similar to the arc
ABC^ but having a radius fr. As the matter of
each elementary sector can be considered concen-
trated in its centre of gravity, it follows that the
centre of gravity of the whole sector is the same as
the centre of gravity of said arc. Hence, applying
to our case the result of No. Ill,
_ ^ X^c _2r c _2r sin &
'"larcXB^"" 3" arc^^(7"~¥ ' t? "
114. Centre of gravity of a parabolic area.
In the parabola, whose equation is 2/' = 2^rr, we
have for an area comprised between the curve and
a double ordinate,
dr)-2ydx =2 V^.x^dx, xdv=2 s/^,^dx;
hence
INFINITESIMAL CALCULUS. 261
J X dx
^0 = 7^ —
J X dx
Let a be the terminal abscissa of the area in ques-
tion. Integraring between the limits ^ = and
;r = a, we shall have
a value independent of the parameter of the curve.
115. Centre of gramty of a paraboloid of revo-
lution^ In this case
dti = Tty^dx, xdv = Tty^xdx^
that is,
dxi = 27r. pxdxy xdv = 2;r .pctfdx.
Hence
Jx^dx
f xdx
and integrating from a?=:0 to a? = a, and reducing,
a value independent of the parameter of the curve.
116. Centre of gramty of ^^^^
a right pyramid. Let (Fig.
48) be the vertex, and OA = h
the height of the pyramid, o'
Calling & the base, any section
parallel to it at a distance
a?
X from the vertex, will be = J ^ , and its infini-
"hc^d'X
tesimal volume will be ,, . Hence
262 INFINITESIMAL CALCULUS.
dv = Tt ixfdx^ and xdv = Ti sfdx /
therefore
fafdx
and integrating from a? = to a? = A, and reducing,
For the frustum of a pyramid^ the integral is to
be taken from x^=^h' to a; = h. This would give
If the base of the pyramid becomes a circle, the
above equations will give the centre of gravity of
the cone, and of the frustum of a cone.
117, Centre of gravity of a spherical zone. We
have from Geometry
dv=27tr.dXy and xdv = 27:r,xdx;
whence
f xdx
J ^^
or, integrating from oj = a to a; == r, and reducing.
If a = 0, we have x^^r=z\r for the centre of grav-
ity of the surface of a liemisphere.
INFINITESIMAL CALCULUS. 263
118. Centre of gravity of a spherical sector.
The spherical sector may be conceived as a sum of
equal infinitesimal pyramids having a common
vertex in the "centre of the sphere, and an infini-
tesimal base on its surface. Each such pyramid
has its centre of gravity at a distance fr from the
centre of the sphere ; so that we may consider the
surface passing tijrough all sucli centres of gravity
as forming a spherical zone with a radius fr.
Hence the centre of gravity of the whole spherical
sector will be found by substituting fr and |a for
r and a in the n?sult of No. 1 17. Therefore
^0 = f (^ + «)•
If a = 0, then Xq = ^ will be the distance of the
centre of gravity of a solid hemisphere.
Moment of Inertia.
119. Let OJl — r.(Fig. 49) be the radius of a
cylinder having a mass m. If its axis be hori-
zontal, and a weight P be attached
to the cylinder by a string wrapped ^^'^
around its surface, the cylinder will
be caused to revolve about its axis.
Let -3T be the angular velocity im-
parted to every element of the mass
at the time t. Then any element
d/m^ whose distance from the axis is
d&
X = 0(7, will have a velocity x -jr ,
dd^
and its quantity of movement will be dm . x -^
264 INFINITESIMAL CALCULUS.
The accelerating action of P on such an element
will be, at this instant, dm.x -^ , and its moment
with respect to the axis will be dm. a? -nr . Hence
the moment of the action of P on the whole mass
m will be the sum of all such moments (No. 106),
that is,
^ {x'dm + x''dm' + x'^dm" +...).
E7 / of dm.
or, briefly,
dP
But the moment of the action of P is also ex-
pressed by Pt ; hence
d^& r ,. ^ , d?d^ Pr
-TTii- / x^dm = PTy and -r^ = —.
dt'J dt^ y^^'^m
Tlie quantity / x^dm is called the moment of
inertia of the mass m with respect to the axis
passing through its centre of gravity. If by per-
forming the integration we find
/
a?dm = mV^ (1)
k is called the radius of gyration^ inasmuch as the
movement will be the same as if the whole mass m
were collected together at the distance Ic from the
axis of rotation.
120. It is to be remarked that the moment of
inertia varies in the same body according to tne
INFINITESIMAL CALCULUS.
266
Fig, 50
positioiL of the axis of rotation. To investigate the
law of its variation, let ABCD (Pig. 50) be a section
of the mass 77^ by a plane
perpendicular to the axis of q
rotation, the i)oint where
the axis is cut by this
plane, and O the point where
a parallel axis passing
through the centre of grav-
ity of the mass is cut by the
same plane. Considering an element d/m of this
mass occupying any position JS?, and denoting OE
by Xy QE by 2, and OG by p, the triangle OOE will
give us the equation
x' = f-^-p^-^pz cos OQE,
Substituting in (1), and separating the terms, we
have
mlc^ = / fdmi + / p^dm — 2 / pzdmi cos OQE^
or, since the distance p is constant, and / dm, = w,
m&'= / ^dmi + Wjo' - 2/> J dmi.z cos OG^JS,
But z cos 0(?-fi? = (7j£r= the lever arm of the mass
dm with respect to the axis passing through the
centre of gravity of the body. Hence
/
zdm cos OOE
is the algebraic sum of the moments of all the par-
ticles of the body with respect to the axis passing
through its centre of gravity ; and this sum, by the
^ I
266
INFINITESIMAL CALCULUS.
principle of moments (No. 106) must be =0, be-
cause the moment of their resultant is also = 0.
We have, therefore, simply
mV = / fdm + mp*.
Now / z^dm is the moment of inertia of the mass
m with respect to the axis passing through the
centre of gravity. Denoting it by w;Vj we shall
have
Therefore, tJie moment of inertia of a body with re-
spect to any axis is equal to the moment qf inertia
with respect to a parallel axis through the centre
of gravity of the hody^ plus the mass of the body
into the sqicare qf the distance between the ttoo a^es.
131. Moment of iner tia of a rectangle. Let PQ
(Pig. 61) be an axis passing through the centre of
gravity of a rectangle, and
lying in the plane of the
rectangle perjoendicularly to
its length AB. Let m be the
mass of the rectangle, and
AB = 2a. Tlie iniinitesimal
element CD = dm placed at any distance x from
the axis, will be found by the proportion
2a : dx::m : dm; or dm=i— dx.
Substituting in (1), we have
Fig. 51
p
G
•
•
A qI
D B
INFINITESIMAL CALCULUS.
267
and integratmg from a? = — a to a? = a, and reduc-
ing,
m1^=zvi
Hence the radius of gyration is here Jc =
a
V3
This result is independent of the altitude of the
rectangle. Hence considering the straight line AB
as a rectangle having an infinitesimal altitude, its
a'
moment of inertia will also he m -^ ^ m denoting
the mass of the line.
122. Moment of inertia of a circle, when the
axis coincides with a diameter AB,
Let OC=r (Fig. 52) be the radius of
the circle, CD = dm an element of
its area parallel to the axis, 0^= x
its distance from the centre O. We
shall have
TTT* : 2pdx ::m: dm^ or dm = — j ydx.
Substituting in (1) and remembering
that 2/ is = Vr' — x% we have
By formula (33) we have
f(r'-x^Yx'dx= ~|(r'-aj')* + ^y^C^"-
a^)^dxj
and by formula (34),
268
INFINITESIMAL CALCULUS.
X
r
r{f^a?^dx = h i/7^^^ + 2 si^"'
Substituting, and taking the integral from x
to ir = r, we have
4
Hence the radius of gyration is here * = 2 •
= — r
Tig. 53
123. Moment of inertia of a circle^ when the
axis through the centre is perpendictUar to tJie
plane of the circle. With a
radius 00 =x (Pig. 53) describe
a circle, and give to its circum-
ference a width dx. Then 27:a^dx
will represent an element of the
area, and thus
;rr* : 27rxdx::m : dm;
or
dm = — T" xdx.
Substituting in (1), and integrating from rr = to
a? = r, we have
This formula is independent of the thickness of the
circular plate ; hence it will be true for any thick-
ness. It therefore expresses the moment of inertia
of a solid cylinder of any length, revolving about
its axis.
INFINITESIMAL CALCULUS.
269
124. Moment of inertia of a circular ring with
respect to an axis perpendicular to its plane. Let
r and r' be the extreme radii of the ring, and m its
mass. Taking x between r and r', we have
;r(r* — r") : 27rxdx::m : dm; and dm = -5 75 a?e?a;.
Substituting in (1), and integrating from x=:r^ to
x = r, we find
,, * m r* — r'* r* + r'*
mk* = -^ -5 7i = m — ^
2 r* — r ' 2
This formula, being independent of the thickness
of the ring, will be true for a hollow cylinder of
any length.
135. Moment of inertia of a cylinder with re-
spect to an axis perpendicular to the a^is of the
cylinder. Let the axis PQ
(Pig. 54) be taken tlirough
the centre of gravity of the
cylinder, and let BD be
an element perpendicular to
the axis of the cylinder, at
a distance 0(7= x from the
axis of rotation. Let r be
the radius of the cylinder,
and // the mass of the element BD. The moment
of inertia of this element with respect to one of its
P
B
diameters would be (No. 122) = // j ; but its mo-
ment of inertia with respect to the axis PQ parallel
to that diameter, and placed at a distance OC=zx^
will be /^ ( J- + ^') 5 as we have shown (No. 120).
270 INFIXITESIMAL CALCULUS.
Now, let 2a be the length, and m the mass, of
the cylinder. We shall have
/jl: 7n::dx :2ay or /i = ^ dx;
and therefore the moment of inertia of the whole
cylinder will be found by integmting the expres-
sion X- I J- +^j ^ between x= -^ a and ic = a.
Integrating, and reducing, we shall find
mk' = m (^ + 1).
136. Moment of inertia of a sphere. Let the
axis pass through the centre of the sphere. Let r
be the radius of the sphere and fi the mass of an
elementary segment perpendicular to the * axis,
placed at a distance x from the centre, and having
a radius y= Vr^ -- x\ The moment of inertia of
this element with respect to the axis is (No. 123)
/i| , or I (r'-rr>).
Now, if m be the mass of the sphere, we have
fi:m:\ izy ax : -g- ;
whence
and substituting this value of // in the above ex-
pression, we shall have for the moment of inertia
b
INFINITESIMAL CALCULUS.
of the elementai'y segment
Integrating this from a: =: — r fo x=^ri
we get, after reduction,
for the moment of inertia of the whole
sphere.
137. Problem. As an application of
the theory of the moments of inertia,
let R and r (Pig. 55) be the radii of two
solid cylinders having the same hori-
zontal axis, M being the mass of the
larger and m that of the smaller ; and
let tlie weights p and P be applied to
them respectively by a thread wrapped
on their surfaces. If the two cylinders
are so connected that they must rotate
together, and if the weighty acts in a
direction opposite to that of P, what
will be the movement of these weights
after a time fi
Solution. Adapting to our case the
formula (No. 119)
^ - _^_
'^ ~ f^dm'
and reflecting that the moments of inertia of the
272
INFINITESIMAL CALCULUS.
two solid cylinders are M -^ and m ^ respect-
ively, and that the moments of the accelerating
fonses are Pr and pB^ we shall have
tP^_ Pr-Rp ,, Pr-Bp
2 2
Let now M' and m' be the masses of the weights
P and p respectively. Then g being the action of
gravity^ we have P = M'g, p = m'g. And there-
fore
dC ~^^ Mie-mr" '
and integrating from < = to ^ = ^,
d» _ M't - Rm' « _ ^ 3f V - Rm'
dt ""^^^ Mie-mr' ' ^-^^ MR-mr' '
Ourvilinear Movement.
188. A free point M (Fig. 56) cannot move in a
curve, unless its direc-
tion be continually ng.se
changed by an action pro-
ceeding from some other
direction. Let us con-
ceive this action decom-
posed into two, X and
Ty respectively parallel
to the co-ordinate axes
OX and Y. Draw MJN'
normal, and JfT^ tangent o
to the curve at the point Jf. Resolve X into MT
INFINITESIMAL CALCULUS. 273
tangential, and Mp normal, also Y into MT^ tan-
gential, and Mq normal to the curve. Then the
forces X and Y may be replaced by a tangential
force T=^ MT — MT\ and by a normal^ or centri-
petal force N = J/p + Mq.
Now, calling (? the angle that the element ds of
the curve at M makes with the axis of abscissas,
we have
MT:= Xcos », MT = r sin *,
Mp=:X sin &j Mq = Y cos ^.
Hence
T=Xcos»-Ysm^,
i\r=Xsini? + rcos(?.
But, according to our usual notation,
dt' ' d^'
. t, dy o dx ,
sm I? = :^ , cos ^ = ^y- ;
ds ds
Therefore
dP ds df ds '
The first of these equations can be reduced to
^_ d {da? + dy') _ d {ds') _ 2dsd's _ d's
~ dt' .2ds "^ d^t' . 2ds "" ^tZ^d'/' eZ^' *
The second, being multiplied and divided by ds\
becomes
274 INFINITESIMAL CALCULUS.
ds* d^ydx — d^xdy .
and thi8, according to the remark made by us on
the expression for the i-adius of curvature when t is
the independent variable (No. 46), will become
Such is the expression of the centripetal accelera-
tion. Hence the centripetal force is equal to the
product of the mass into the square of the tangent
tial velocity divided by the radius of curvature
The moving point, while obeying the centripetal
action, always keeps its tendency to follow a
straight line, that is, the tangential direction, and
thus to recede from the centre of curvature. This
centrifugal tendency, so far as counteracted and
thwarted by the centripetal action, is usually called
the centrifugal for ce^ and its intensity is measured
by that of the action by which it is thwarted.
Hence the centrifugal force is equal to the centri-
petal, and directly opposed to it. The centripetal
and centrifugal forces are commonly called central
forces.
139. Let a point M roll down a curve OMC
(Pig. 57) under the action of gravity. Let OX and
Z be the co-ordinate axes, and let the ordinates
downward be positive. When the moving point
has reached any position M, the action of gravity
MOj by which its movement is accelerated, may be
decomposed into MJV normal, and MT tangential
to the curve. The first will be destroyed by the
Infinitesimal calculus.
275
Fig, 57
resistance of the curve, whicli we assume to be in-
variably fixed, and wliicli is
thus playing the part of a
centripetal force. The sec- ^
ond will have its whole effect.
Let d^ be the angle that the
curve at M makea with the \
axis OX Then
MT=g^m»^g^,
or
d^s _ dy
dP~^ di'
Hence
^ds d^
dt dt
= 2gdy, and (J^ = 2gy.
This velocity is the velocity due to the height y
(No. 100). Therefore the velocity acquired by a
body rolling down a curve under the action of
gravity, is equal to that which it acquires by falling
freely through the same vertical height.
This result is true not only when g is constant,
but also when g varies according to a fixed law;
for, even in this case, g may be regarded as con-
stunt from element to element, inasmuch as the
same law of variation applies to the elements of the
curve and to those of rlie vertical line. Hence a
body falling toward the suu on a spiral line will
have the same final velocity as though it had fallen
directly towards its centre.
130: The simple pendulum. A material point
276
INFINITSaiMAL CALCULUS:
A (Fig. 08) suspended from a horizontal axis by
a rigid line AO without
weight, and free to oscil-
late about that axis, con-
stitutes a simple pendu-
lum.
Let AO = ?, the angle
AOC = a, and the angle
MOC = *. When the
point A under the action
of gravity reaches the point
Jf, it will have acquired a
ds
velocity ^ = V^^ (No. 129), y being = DE. But
d8 = ld^^ and y — OE— OB = I (cos * — cos a) ;
therefore
Z ^ = ^2gl (cos & — cos a) ;
whence
dt
-yw
d&
2^7 i^cos^ — cosa
From Maclaurin's formula we have
cos t/ = 1 — TT +
2 * 2.3.4
cosa=l-|' + 2;|^-
• • •
• «
hence, if the arc J. (7 be small enough to allow us
to neglect all the terms of the series after the sec-
ond, we shall have cos t? — cos a:=.\((;f — #•), and
INFINITESIMAL CALCULUS.
277
dt
~ y g' V^r^&' '
Integrating from «>— — afcot? = a, we shall have
t
-Vj--
This is the time of one excursion, or semi- oscilla-
tion, of a simple pendulum, when the amplitude of
the excursion is small; that is, not exceeding 9 or
10 degrees.
131. Planetary orbits. By the first of Kepler's
laws, the orbits of planets are ellipses, of which
one focus is occupied by the
sun. Fig.J9
Let PMA (Fig. 69) be a - ^^
planetary orbit, and JF the /'''^ I^^M
focus occupied by the sun.
Let the planet, at a given PI
instant dt, be moving from
M towards the perihelion P.
Make FM=:p^ the angle
MFA=v, OQ = x, MQ^y, OF^c,
and let